Nous avons le plaisir de vous convier à la défense de thèse de Madame Hélène Vanvinckenroye, qui se déroulera le lundi 17 septembre à 16h (TP64, dans la nouvelle annexe du bâtiment B52). Cette recherche doctorale s'intitule "The first-passage time as an analysis tool for the reliability of stochastic oscillators" et a été réalisée sous la direction du professeur Vincent Denoël.
Because some systems spend most of their time in transient regimes, it is important to focus on a relevant representation of their transient response. The question "How much time ?" is addressed in this work through the first-passage problem. The first-passage time is the time required for a system, leaving a given initial configuration, to reach a certain state for the first time. When the excitation is a stochastic process, the first-passage time is a random variable and the determination of its statistics is an attractive approach for assessing the reliability of a transient system exposed to uncertainty. Although the number of engineering results seen from this angle is very limited today, the first-passage problem has been widely studied in physics and mathematics and presents a high potential for a wide range of engineering problems.
Since many physical problems can be described by a stochastic Mathieu equation, this work provides a frame for the first-passage time of this category of oscillators as an analysis tool in engineering applications.
The first step is the determination of a closed-form expression for the average first-passage time of the linear, undamped Mathieu oscillator under parametric and external white noise excitations. Given by the solution of the Pontryagin equation, the approximate expression is obtained using an asymptotic expansion. The solution highlights the groups of parameters influencing the first-passage time which is presented in a universal map. Three regimes –the incubation, additive and multiplicative regimes– are identified in the map with their typical features.
Next, the complexity of the model is progressively increased, considering for example a damped oscillator or the variance of the first-passage time. The features of the three regimes are re-identified and a new map is determined.
An attempt at fitting our simple model to the complex dynamics of a tower crane oscillating in a turbulent wind flow has proven very satisfactory. This indicates that the three elementary regimes (incubation, additive and multiplicative) are also present in other problems than those circumscribed by the hypotheses of the Mathieu oscillator.
The first-passage maps are calculated using an appropriate algorithm and it is shown that there exists an equivalent linear Mathieu oscillator so that the theoretical model may be used to understand and predict the tower crane behavior. The identification of this map with the analytical model developed before, as well as the observation of the three regimes, serves as a demonstration of the applicability of the first-passage time as an identification tool or a reliability assessment tool in engineering applications.
Finally, a Galerkin scheme is developed to provide a robust and versatile method of resolution of the Backward-Kolmogorov equation governing the first-passage time complete distribution of nonlinear systems under evolutionary excitation.